1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

348 7. THE REDUCED DISTANCE

We now rewrite the £-Jacobi equation in a more natural way in view

of the space-time geometry associated to the Ricci flow. Consider the

quantity Re g(T) (Y). The time-dependent symmetric 2-tensor Re g(T) is de-

fined on all of M whereas Y is a vector field along the path "( ( T) in M.

In local coordinates, Re g(T) (Y)i = gij Rjkyk, so actually we are consid-

ering Re as a (1, 1)-tensor. Caveat: When we take the time-derivative

of Re, we consider it as a (2, 0)-tensor and then raise an index to get a

(1, 1)-tensor! By \7 x [Re (Y)] (To) we simply mean the covariant derivative

along "( ( T) of the vector field Reg( 70 ) (Y ( "! ( T))) at T = To. In this respect

the vector field Reg( ,,. 0 ) (Y ('y ( T))) along "! ( T) should be distinguished from

Rcg(T) (Y ('y (T))), where in the latter case the Ricci tensor depends on time.

Combining the equations^14

Dd~ [Rcg( 7 )(Y)] = (:TRc) (Y)+(\lxRc)(Y)+Rc(\lxY)-2Rc^2 (Y)

and

D d~ (\7~

7

) x) = \lx (\lyX) + (:T \7) Y x

= \7 x (\lyX) + (\ly Re) (X)

• (\7 x Re) (Y) - (\7 Re) ( 1;, 1°) ,

where (\7 Re) ( 1;, 1) ( Z) ~ (\7 Re) ( Z, Y, X) , and commuting derivatives,

we have

(7.123) D ..<L (Re (Y) + \ly X)

d-r

= ( :T Re) (Y) + 2 (\7 x Re) (Y) + Re (\7 x Y) - 2 Re^2 (Y).

• \ly (\7 xX) + R (X, Y) X + (\ly Re) (X) - (\7 Re) (1;, 1°) ,

where we used

\lx (\lyX) = \ly (\lxX) + R (X, Y) X.

Substituting the £-geodesic equation

1 1

0 = \7 x X - :2 \7 R + 2 Re ( X) +

2

T X

in (7.123) and adding this to

D d (-~Y) = -

1

-Y -~\lxY

dr 2T 2T2 2T '

(^14) In accordance with the caveat above, /Jr Re denotes the derivative of Re as a (2, 0)-
tensor and Ur Re) (Y)' ~ gij Ur Rc)jk yk_